Inductive and Deductive Reasoning

Reasoning is used in mathematics, science, and everyday life. When you reach a conclusion, called a generalization, based on several observations, you are using inductive reasoning. Such a generalization is not always true. If a counterexample is found, then the generalization is proved to be false.

Example of Inductive reasoning: You go to the school library every day for a week to do research for a paper. You notice that the computer you plan to use is not in use at 3:00 each day.

a. Using inductive reasoning, what would you conclude about the computer? From your observations, you conclude that the computer is available every weekday at 3:00.

B. What counterexample would show that your generalization is false? A counterexample would be to find the computer in use one day at 3:00.

Using inductive reasoning, name the next three numbers of the sequence 1, 5, 9, 13, 17, 21……. 25, 29, 33

When you use facts, definitions, rules, or properties to reach a conclusion, you are using deductive reasoning. A conclusion reached this way is proved to be true.

Deductive reasoning uses if- then statements. The if part is called the hypothesis and the then part is called the conclusion.

When deductive reasoning has been used to prove an if- then statement then the fact that the hypothesis is true implies that the conclusion is true.

Use of if-then statement During the last week of a class, your teacher tells you that if you receive an A on the final exam, you will have earned an A average in the course.

Identify the hypothesis and the conclusion of the if-then statement. The hypothesis is “you receive an A on the final exam.” The conclusion is “you will have earned an A average in the course.”

Suppose you receive an A on the final exam. What will your final grade be? You can conclude that you have earned an A in the course.

Deductive Reasoning in Mathematics To show that the statement “If a, b, and c, are real numbers, then (a+b)+c = (c+b)+a” is true by deductive reasoning.

You can write each step and justify it using the properties of addition. (a+b)+c = c+(a+b) Commutative property. = c+(b+a) Commutative =(c+b)+a Associative

Get with your group and discuss if the reasoning is inductive or deductive and be able to explain your answer!

1. The tenth number in the list 1, 4, 9, 16, 25,….is 100. Inductive reasoning; a conclusion is reached based on observation of a pattern.

2. You know that in your neighborhood, if it is Sunday, then no mail is delivered. It is Sunday, so you conclude that the mail will not be delivered. Deductive reasoning; a conclusion is reached because the hypothesis is true.

3. If the last digit of a number is 2, then the number is divisible by 2. You conclude that 98,765,432, is divisible by 2. Deductive reasoning; a conclusion is reached because the hypothesis is true.

4. You notice that for several values of x the value of x to the second power is greater than x. You conclude that the square of a number is greater than the number itself. Inductive reasoning; a conclusion is reached based on observation of a pattern.

5. Find a counterexample to show that the conclusion in #4 is false. Example: The square of 1 is equal to itself.

6. Give an example of inductive reasoning and an example of deductive reasoning. Do not use any examples already given! Be prepared to share with the class.

Person B share with the class. Does everyone agree with the answers?

7. Name the next three numbers of the sequence 0, 3, 6, 9, 12, 15, 18,….. 21, 24, 27

Now, with your group: identify the hypothesis and the conclusion of the if-then statement. 8. If you add two odd numbers, then the answer will be an even number.

Hypothesis: “you add two odd numbers” Conclusion: “the answer will be an even number.”

9. If you are in Minnesota in January; then you will be cold. Hypothesis: “you are in Minnesota in January” Conclusion: “you will be cold”

10. Use deductive reasoning and the properties of addition to show that if z is a real number, then (z +2)+(-2) = z

(z+2)+(-2) = z + (2 +(-2)) Associative property = z + 0 Inverse property = z (identity property)

Scott tells Michael “If you catch two fly balls during one game, then I will pay you $10.” Michael catches two fly balls during one game. Identify the hypothesis and conclusion of the if-then statement.

Hypothesis: you catch two fly balls during one game. Conclusion: I will pay you $10.

Use deductive reasoning to determine whether Michael gets $10. Michael catches two fly balls during one game, so he will get paid $10.

Sequences Is an ordered list of numbers. Each number in a sequence is called a term.

Arithmetic sequence Each term is found by adding the same number to the previous term.

Examples: 8, 11, 14, 17, 20…. Adding 3

Geometric Each term is found by multiplying the previous term by the same number.

Example: 3, 6, 12, 24, 48…. Multiplying by 2